#460123

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,16)(8,20,22,29)(9,23)(10,14)(11,15)(12,18)(13,21,30,24)(17,26,33,36)(19,27)(25,28)(31,34,35,32);(1,2)(3,4)(5,7,9,18,11,19,10,25)(6,12,14,16,15,28,23,27)(8,17,24,34,22,33,21,32)(13,26,20,31,30,36,29,35);(2,4)(8,21,22,24)(13,29,30,20)(17,32,33,34)(26,35,36,31);(5,8,9,24,11,22,10,21)(6,13,14,20,15,30,23,29)(7,17,18,34,19,33,25,32)(12,26,16,31,28,36,27,35)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367319);